1.Introduction


I show that there is a parameter solution for 6.7.7 by Gloden's theorem.



2. Gloden's theorem

             A1 + A2 + A3 + A4 + A6 + A7 = 0

             A13+ A23+ A33+ A43+ A63+ A73 = 0

             A15+ A25+ A35+ A45+ A65+ A75 = 0

             There is a parametric solution of above simultaneous Diophantine equation.

             A1=-(f2-f*k+k2)*(-3*k*f2+k3+f3)
             A2=-(k-f)*(f+k)*(f2-3*f*k+k2)*f
             A3=(-f+2*k)*(-f2-f*k+k2)*k*f
             A4=(k-f)*(k-2*f)*(-f2+f*k+k2)*k
             A5=(k-f)*(f4-2*k*f3-k2*f2+k4)
             A6=-(k4-2*f*k3-k2*f2+4*k*f3-f4)*k
             A7=-(k4-5*k2*f2+4*k*f3-f4)*f


             For example,take (f,k)=(3,1) then

             -7  + 24  + 33  + (-50)  + (-38)  + (-13)  + 51 = 0

             -73 + 243 + 333 + (-50)3 + (-38)3 + (-13)3 + 513 = 0

             -75 + 245 + 335 + (-50)5 + (-38)5 + (-13)5 + 515 = 0
   
3. Solution of 6.7.7

            f=(A1x+1)6-(A1x-1)6+(A2x+1)6-(A2x-1)6+(A3x+1)6-(A3x-1)6
             +(A4x+1)6-(A4x-1)6+(A5x+1)6-(A5x-1)6+(A6x+1)6-(A6x-1)6+(A7x+1)6-(A7x-1)6


            We know that above polinomial becomes 0.

            For example,take (f,k)=(3,1) then

           (-7x+1)6-(-7x-1)6+(24x+1)6-(24x-1)6+(33x+1)6-(33x-1)6+(-50x+1)6-(-50x-1)6
          +(-38x+1)6-(-38x-1)6+(-13x+1)6-(-13x-1)6+(51x+1)6-(51x-1)6 = 0

          Take x=1,then

          (-6)6+ 256+ 346+ (-49)6+ (-37)6+ (-12)6+ 526 = 86+ (-23)6+ (-32)6+ 516+ 396+ 146+ (-50)6

  



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